In an isoceles triangle ABC, the acute angle BAC is `36^@` Point D lies on `bar (AC)` such that `bar (BD)` is the angle bisector of `angle ABC`. Find `bar (AB)/bar (BC)`.

Options

A. A)`sqrt5 : 2`
B. B)`(sqrt5 - 1)/2`
C. C)`(1 + sqrt5)/2`
D. D)none of these

Answer

To solve the problem, we need to find the ratio of the sides \( \frac{AB}{BC} \) in the isosceles triangle \( ABC \) where \( \angle BAC = 36^\circ \) and \( BD \) is the angle bisector of \( \angle ABC \).

### Step-by-step Solution:

1. **Identify the Angles in Triangle ABC**:
- Since triangle \( ABC \) is isosceles with \( AB = AC \), we know that \( \angle ABC = \angle ACB \).
- Let \( \angle ABC = \angle ACB = x \).
- Using the triangle angle sum property, we have:
\[ \angle BAC + \angle ABC + \angle ACB = 180^\circ \]
\[ 36^\circ + x + x = 180^\circ \]
\[ 2x = 180^\circ - 36^\circ = 144^\circ \]
\[ x = 72^\circ \]
- Therefore, \( \angle ABC = \angle ACB = 72^\circ \).

**Hint**: Remember that the angles in a triangle sum up to \( 180^\circ \).

2. **Apply the Angle Bisector Theorem**:
- Since \( BD \) is the angle bisector of \( \angle ABC \), it divides \( \angle ABC \) into two equal parts:
\[ \angle ABD = \angle DBC = \frac{72^\circ}{2} = 36^\circ \]

3. **Use the Sine Rule**:
- According to the sine rule in triangle \( ABC \):
\[ \frac{AB}{BC} = \frac{\sin(\angle ACB)}{\sin(\angle BAC)} \]
- Substituting the angles we found:
\[ \frac{AB}{BC} = \frac{\sin(72^\circ)}{\sin(36^\circ)} \]

4. **Calculate the Values**:
- Using known values of sine:
- \( \sin(72^\circ) = \frac{\sqrt{5} + 1}{4} \)
- \( \sin(36^\circ) = \frac{\sqrt{5} - 1}{4} \)
- Therefore:
\[ \frac{AB}{BC} = \frac{\frac{\sqrt{5} + 1}{4}}{\frac{\sqrt{5} - 1}{4}} = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \]

5. **Simplify the Ratio**:
- To simplify \( \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \), we can multiply the numerator and denominator by \( \sqrt{5} + 1 \):
\[ \frac{(\sqrt{5} + 1)^2}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{5 + 2\sqrt{5} + 1}{5 - 1} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} \]

6. **Final Result**:
- Thus, the ratio \( \frac{AB}{BC} \) is:
\[ \frac{AB}{BC} = \frac{3 + \sqrt{5}}{2} \]

### Final Answer:
\[ \frac{AB}{BC} = \frac{3 + \sqrt{5}}{2} \]